The DOI matches its Crossref record; 64 of the 64 references it cites were found in a registry or an index.
A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics
Open accessAbstract
Abstract Bifurcations can cause dynamical systems with slowly varying parameters to transition to far-away attractors. The terms “critical transition” or “tipping point” have been used to describe this situation. Critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance. The main goal of this paper is to give an overview which standard mathematical theories can be applied to critical transitions. We shall focus on early-warning signs that have been suggested to predict critical transitions and point out what mathematical theory can provide in this context. Starting from classical bifurcation theory and incorporating multiple time scale dynamics one can give a detailed analysis of local bifurcations that induce critical transitions. We suggest that the mathematical theory of fast–slow systems provides a natural definition of critical transitions. Since noise often plays a crucial role near critical transitions the next step is to consider stochastic fast–slow systems. The interplay between sample path techniques, partial differential equations and random dynamical systems is highlighted. Each viewpoint provides potential early-warning signs for critical transitions. Since increasing variance has been suggested as an early-warning sign we examine it in the context of normal forms analytically, numerically and geometrically; we also consider autocorrelation numerically. Hence we demonstrate the applicability of early-warning signs for generic models. We end with suggestions for future directions of the theory.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
Guckenheimer, J., & Holmes, P. (1983) · Applied Mathematical Sciences · doi:10.1007/978-1-4612-1140-2 · Cited by 16,144
The Fokker-Planck Equation: Methods of Solution and Applications
Risken, H. (1996) · Springer Series in Synergetics · doi:10.1007/978-3-642-61544-3 · Cited by 9,056
Catastrophic shifts in ecosystems
Scheffer, M., Carpenter, S., Foley, J. A., Folke, C., & Walker, B. (2001) · Nature · doi:10.1038/35098000 · Cited by 7,510
Elements of Applied Bifurcation Theory
Kuznetsov, Y. A. (2004) · Applied Mathematical Sciences · doi:10.1007/978-1-4757-3978-7 · Cited by 5,564
Numerical solution of stochastic differential equations
(2016) · doi:10.5194/gmd-2016-45-rc1 · Cited by 5,002
Early-warning signals for critical transitions
Scheffer, M., Bascompte, J., Brock, W. A., Brovkin, V., Carpenter, S. R., Dakos, V., Held, H., van Nes, E. H., Rietkerk, M., & Sugihara, G. (2009) · Nature · doi:10.1038/nature08227 · Cited by 4,854
(1998) · doi:10.1142/3481 · Cited by 4,436
Tipping elements in the Earth’s climate system
Lenton, T. M., Held, H., Kriegler, E., Hall, J. W., Lucht, W., Rahmstorf, S., & Schellnhuber, H. J. (2008) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0705414105 · Cited by 3,982
An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations
Higham, D. J. (2001) · SIAM Review · doi:10.1137/s0036144500378302 · Cited by 3,154
Stochastic Differential Equations
Øksendal, B. (2003) · Universitext · doi:10.1007/978-3-642-14394-6_5 · Cited by 3,081
Cited by
Anticipating Critical Transitions
Scheffer, M., Carpenter, S. R., Lenton, T. M., Bascompte, J., Brock, W., Dakos, V., van de Koppel, J., van de Leemput, I. A., Levin, S. A., van Nes, E. H., et al. (2012) · Science · doi:10.1126/science.1225244 · Cited by 2,208
Exceeding 1.5°C global warming could trigger multiple climate tipping points
Armstrong McKay, D. I., Staal, A., Abrams, J. F., Winkelmann, R., Sakschewski, B., Loriani, S., Fetzer, I., Cornell, S. E., Rockström, J., & Lenton, T. M. (2022) · Science · doi:10.1126/science.abn7950 · Cited by 1,931
Ashwin, P., Wieczorek, S., Vitolo, R., & Cox, P. (2012) · Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences · doi:10.1098/rsta.2011.0306 · Cited by 684
Resilience indicators: prospects and limitations for early warnings of regime shifts
Dakos, V., Carpenter, S. R., van Nes, E. H., & Scheffer, M. (2015) · Philosophical Transactions of the Royal Society B: Biological Sciences · doi:10.1098/rstb.2013.0263 · Cited by 578
Climate change, ecosystems and abrupt change: science priorities
Turner, M. G., Calder, W. J., Cumming, G. S., Hughes, T. P., Jentsch, A., LaDeau, S. L., Lenton, T. M., Shuman, B. N., Turetsky, M. R., Ratajczak, Z., et al. (2020) · Philosophical Transactions of the Royal Society B: Biological Sciences · doi:10.1098/rstb.2019.0105 · Cited by 382
Sivapalan, M., & Blöschl, G. (2015) · Water Resources Research · doi:10.1002/2015wr017896 · Cited by 353
Physics-based early warning signal shows that AMOC is on tipping course
van Westen, R. M., Kliphuis, M., & Dijkstra, H. A. (2024) · Science Advances · doi:10.1126/sciadv.adk1189 · Cited by 293
Lenton, T. M. (2013) · Annual Review of Environment and Resources · doi:10.1146/annurev-environ-102511-084654 · Cited by 277
Early warning signals also precede non‐catastrophic transitions
Kéfi, S., Dakos, V., Scheffer, M., Van Nes, E. H., & Rietkerk, M. (2013) · Oikos · doi:10.1111/j.1600-0706.2012.20838.x · Cited by 264
Dai, L., Korolev, K. S., & Gore, J. (2015) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1418415112 · Cited by 78
Metadata
- Authors
- Kuehn, C.
- Venue
- Physica D: Nonlinear Phenomena
- Publisher
- Elsevier BV
- Published
- 2011-06
- Volume
- 240
- Issue
- 12
- Pages
- 1020-1035
- ISSN
- 0167-2789 (ISSN-L)
- OpenAlex
- W2168773352
- Cited by
- 411
- References
- 64
- FWCI
- 18.26
- Type
- journal article
- Language
- en
- Open access
- green
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
